The -basis and implicitization of a rational parametric surface

• Autores: Falai Chen, David Cox, Yang Liu
• Localización: Journal of symbolic computation, ISSN 0747-7171, Vol. 39, Nº 6, 2005, págs. 689-706
• Idioma: inglés
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• Resumen

  • The concept of a µ-basis was introduced in the case of parametrized curves in 1998 and generalized to the case of rational ruled surfaces in 2001. The µ-basis can be used to recover the parametric equation as well as to derive the implicit equation of a rational curve or surface. Furthermore, it can be used for surface reparametrization and computation of singular points. In this paper, we generalize the notion of a µ-basis to an arbitrary rational parametric surface. We show that: (1) the µ-basis of a rational surface always exists, the geometric significance of which is that any rational surface can be expressed as the intersection of three moving planes without extraneous factors; (2) the µ-basis is in fact a basis of the moving plane module of the rational surface; and (3) the µ-basis is a basis of the corresponding moving surface ideal of the rational surface when the base points are local complete intersections. As a by-product, a new algorithm is presented for computing the implicit equation of a rational surface from the µ-basis. Examples provide evidence that the new algorithm is superior than the traditional algorithm based on direct computation of a Gröbner basis. Problems for further research are also discussed.